Deforming the Lie algebra of vector fields on $S^1$ inside the Poisson algebra on ${\dot{T}}^{\ast }{S}^1$

Abstract

We study deformations of the standard embedding of the Lie algebra $\Vect(S^1)$ of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle $T^{\ast }{S}^1$ (with respect to the Poisson bracket). We consider two analogous but different problems: (a) formal deformations of the standard embedding of $\Vect(S^1)$ into the Lie algebra of functions on $\dot T^*S^1:=T^*S^1\setminusS^1$ which are Laurent polynomials on fibers, and (b) polynomial deformations of the $\Vect(S^1)$ subalgebra inside the Lie algebra of formal Laurent series on ${\dot{T}}^{\ast }{S}^1$.

19 pages, LaTex

Cited by in corpus (12)

• Cohomology of the Lie Superalgebra of Contact Vector Fields on ${\mathbb{R}}^{1|1}$ and Deformations of the Superspace of Symbols
• Deformations of modules of differential forms
• Deformation of Vect($\mathbb{R})$-Modules of Symbols
• A spinor-like representation of the contact superconformal algebra K'(4)
• Multidimensional integrable systems and deformations of Lie algebra homomorphisms
• Embedding of the Lie superalgebra $D(2,1;\alpha )$ into the Lie superalgebra of pseudodifferential symbols on $S^{1|2}$
• Deformation of sl(2) and osp(1|2)-Modules of Symbols
• On matrix realizations of the contact superconformal algebra ${\hat{K}}^{\prime }(4)$ and the exceptional N = 6 superconformal algebra
• On cohomology of the Lie superalgebra D(2, 1 ; α)
• On matrix realizations of the Lie superalgebra D(2, 1 ; α)
• On representations of the exceptional superconformal algebra $C{K}_6$
• Deforming the Lie Superalgebra $\mathcal{K}(1)$-Modules Of Symbols